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Questions and Answers
What defines a set in mathematics?
What defines a set in mathematics?
- A collection of objects with a defined order.
- An unordered collection of objects. (correct)
- A collection where elements can repeat.
- An ordered collection of objects.
Which notation indicates that an element is not a member of a set?
Which notation indicates that an element is not a member of a set?
- 𝑒 ⊂ 𝑆
- 𝑒 ∉ 𝑆 (correct)
- 𝑒 = 𝑆
- 𝑒 ∈ 𝑆
Which of the following represents an open interval?
Which of the following represents an open interval?
- (𝑎, 𝑏) (correct)
- (𝑎, 𝑏]
- [𝑎, 𝑏]
- [𝑎, 𝑏)
Two sets A and B are considered equal if:
Two sets A and B are considered equal if:
What is the representation of the empty set?
What is the representation of the empty set?
Which set correctly represents the odd positive integers less than 10?
Which set correctly represents the odd positive integers less than 10?
Which of the following correctly represents a closed interval?
Which of the following correctly represents a closed interval?
What is the cardinality of the set 𝑆 = {𝑎, 𝑏, 𝑐, 𝑑}?
What is the cardinality of the set 𝑆 = {𝑎, 𝑏, 𝑐, 𝑑}?
Which of the following sets is an example of an empty set?
Which of the following sets is an example of an empty set?
If 𝐴 = {1, 2, 3, {2,3}, 9}, what is the cardinality of set 𝐴?
If 𝐴 = {1, 2, 3, {2,3}, 9}, what is the cardinality of set 𝐴?
The set of positive integers is categorized as what type of set?
The set of positive integers is categorized as what type of set?
What notation is used to indicate that set 𝐴 is a subset of set 𝐵?
What notation is used to indicate that set 𝐴 is a subset of set 𝐵?
When can two sets A and B be considered equal?
When can two sets A and B be considered equal?
How is a proper subset different from a regular subset?
How is a proper subset different from a regular subset?
What is the cardinality of the empty set?
What is the cardinality of the empty set?
Which of the following statements is true about finite and infinite sets?
Which of the following statements is true about finite and infinite sets?
What does the notation $A \subset B$ represent?
What does the notation $A \subset B$ represent?
How many elements are in the power set of the set $S = {1, 2, 3}$?
How many elements are in the power set of the set $S = {1, 2, 3}$?
Which of the following sets is a power set of $S = {a, b}$?
Which of the following sets is a power set of $S = {a, b}$?
What is the Cartesian product $A \times B$ if $A = {1, 2}$ and $B = {a, b, c}$?
What is the Cartesian product $A \times B$ if $A = {1, 2}$ and $B = {a, b, c}$?
Which statement about the element 3 in the set ${1, 2, 3, 4, 7}$ is true?
Which statement about the element 3 in the set ${1, 2, 3, 4, 7}$ is true?
How many total subsets does the set $B = {0, 3, 5, 7, 9}$ have?
How many total subsets does the set $B = {0, 3, 5, 7, 9}$ have?
If 3 is an element of the set ${1, 2, 1, 3}$, what can we conclude?
If 3 is an element of the set ${1, 2, 1, 3}$, what can we conclude?
What is the universal set in the context of sets A, B, and C defined as above?
What is the universal set in the context of sets A, B, and C defined as above?
What would the set $A \cap C$ yield if $A = {1, 2, 3, 4, 7}$ and $C = {1, 2}$?
What would the set $A \cap C$ yield if $A = {1, 2, 3, 4, 7}$ and $C = {1, 2}$?
What is the intersection of the sets {1, 3, 5} and {1, 2, 3}?
What is the intersection of the sets {1, 3, 5} and {1, 2, 3}?
Which of the following statements is true about disjoint sets?
Which of the following statements is true about disjoint sets?
What does the difference of sets A and B, denoted by A - B, represent?
What does the difference of sets A and B, denoted by A - B, represent?
How is the complement of a set A defined in relation to the universal set U?
How is the complement of a set A defined in relation to the universal set U?
Which of the following correctly evaluates the difference A - B for A = {1, 3, 5} and B = {1, 2, 3}?
Which of the following correctly evaluates the difference A - B for A = {1, 3, 5} and B = {1, 2, 3}?
What is the result of the Cartesian product A × B if A = {1, 2} and B = {a, b, c}?
What is the result of the Cartesian product A × B if A = {1, 2} and B = {a, b, c}?
If A = {1, 3, 5} and B = {1, 2, 3}, what is A ∪ B?
If A = {1, 3, 5} and B = {1, 2, 3}, what is A ∪ B?
What does the intersection of sets A and B, denoted as A ∩ B, signify?
What does the intersection of sets A and B, denoted as A ∩ B, signify?
How many elements are in the Cartesian product A × B if A contains 2 elements and B contains 3 elements?
How many elements are in the Cartesian product A × B if A contains 2 elements and B contains 3 elements?
What is the result of B × A if A = {1, 2} and B = {a, b, c}?
What is the result of B × A if A = {1, 2} and B = {a, b, c}?
If the set A = {1, 2, 3} and the set B = {3, 4, 5}, what is A ∩ B?
If the set A = {1, 2, 3} and the set B = {3, 4, 5}, what is A ∩ B?
Which of the following describes the concept of union of two sets?
Which of the following describes the concept of union of two sets?
In the context of the Cartesian products, what does the notation A1 × A2 × ... × An represent?
In the context of the Cartesian products, what does the notation A1 × A2 × ... × An represent?
If A = {1, 2} and B = {a}, what is A × B?
If A = {1, 2} and B = {a}, what is A × B?
What is the primary distinction between union and intersection of sets?
What is the primary distinction between union and intersection of sets?
Flashcards
Set Definition
Set Definition
A set is an unordered collection of objects. The objects are called elements or members of the set.
Set Element Notation
Set Element Notation
We use '∈' to show an element is in a set and '∉' to show an element is not.
Set Examples
Set Examples
Sets can be displayed using listing or set-builder notation (e.g., {1, 2, 3...99}).
Set Equality
Set Equality
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Empty Set
Empty Set
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Closed Interval
Closed Interval
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Interval Notation
Interval Notation
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Subset
Subset
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Element of a Set
Element of a Set
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Set Notation
Set Notation
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Power Set
Power Set
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Venn Diagram
Venn Diagram
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Universal Set
Universal Set
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Cartesian Product
Cartesian Product
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Ordered Pair
Ordered Pair
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Set Membership
Set Membership
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Symbol: ∈
Symbol: ∈
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Cardinality
Cardinality
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Set Notation for Cardinality
Set Notation for Cardinality
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Proper Subset
Proper Subset
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Infinite Set
Infinite Set
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What does A ⊆ B mean?
What does A ⊆ B mean?
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What does |S| represent?
What does |S| represent?
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Cartesian Product (A x B)
Cartesian Product (A x B)
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Cartesian Product (A x B) Example
Cartesian Product (A x B) Example
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Cartesian Product (B x A)
Cartesian Product (B x A)
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Cartesian Product (n sets)
Cartesian Product (n sets)
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Union (A U B)
Union (A U B)
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Intersection (A ∩ B)
Intersection (A ∩ B)
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Union Example
Union Example
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What is a Set?
What is a Set?
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What is an Element?
What is an Element?
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Set Intersection
Set Intersection
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Disjoint Sets
Disjoint Sets
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Set Difference
Set Difference
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Set Complement
Set Complement
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What is the complement of a set A?
What is the complement of a set A?
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Study Notes
Discrete Mathematics - Basic Structures
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Sets are unordered collections of objects
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The objects in a set are called elements or members
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Sets are denoted using curly brackets {}
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Set notation uses ∈ to denote that an object is an element of a set, and ∉ to denote that an object is not an element
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Examples of sets:
- Set O of odd positive integers less than 10 = {1, 3, 5, 7, 9}
- Set of positive integers less than 100 = {1, 2, 3, ..., 99}
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Set builder notation: another way to define a set
- Example: O = {x|x is an odd positive integer less than 10}
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Examples of standard sets:
- N = {0, 1, 2,...} (natural numbers)
- Z = {..., -2, -1, 0, 1, 2...} (integers)
- Z+ = {1, 2, 3,...} (positive integers)
- Q = {p/q | p ∈ Z, q ∈ Z, and q ≠ 0} (rational numbers)
- R = (real numbers)
- R+ (positive real numbers)
- C (complex numbers)
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Interval Notation - defines ranges of real numbers
- Closed interval [a, b] = {x | a ≤ x ≤ b}
- Open interval (a, b) = {x | a < x < b}
- Half-open intervals:
- [a, b) = {x | a ≤ x < b}
- (a, b] = {x | a < x ≤ b}
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Equal Sets: if A and B are equal if and only if ∀x (x ∈ A ↔ x ∈ B)
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Empty Set: denoted by {} or Ø; has no elements
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Cardinality: the number of distinct elements in a set, denoted by |S|
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Example 1 demonstrates set notation and cardinality
- S = {a, b, c, d} |S| = 4
- A = {1, 2, 3, 7, 9} |A|= 5
- Ø = {} |Ø| = 0
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Example2 demonstrates set notation and cardinality
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Infinite Sets: sets that are not finite
- Z+ = {1, 2, 3...} (positive integers) is an infinite set
Subsets
- Subset (⊆): Set A is a subset of set B if every element of A is also an element of B (written as A ⊆ B).
- A ⊆ B ⇔ ∀ x(x ∈ A → x ∈ B)
- Proper Subset (⊂): If A is a subset of B, but A ≠ B, then A is a proper subset of B (written as A ⊂ B). A proper subset cannot equal the original set.
Set Operations
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Union (∪): The union of sets A and B (A∪B) is the set containing all elements that are in either A, B, or both.
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Intersection (∩): The intersection of A and B (A∩B) is the set containing all elements in both A and B.
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Disjoint Sets: Two sets are disjoint if their intersection is the empty set (A∩B = Ø).
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Sets Difference: A − B = {x | x ∈ A and x ∉ B}
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This gives the elements in A that are not in B.
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Complement (Ā): The complement of set A (Ā) in universal set U is the set of elements in U that are not in A.
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Generalized Unions: Using the notation ⋃i=1nAi\bigcup_{i=1}^{n}{A_i}⋃i=1nAi, to denote the union of several sets.
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Generalized Intersections: Using the notation ⋂i=1nAi\bigcap_{i=1}^{n}{A_i}⋂i=1nAi, to denote the intersection of several sets.
Cartesian Product
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The Cartesian product of sets A and B (A x B), is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B
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Example: Let A = {1, 2} and B = {a, b, c}. A x B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}
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The cardinality of the Cartesian product |A x B| = |A| * |B| = 2 * 3 = 6.
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A × B × C etc...
Set Identities
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These are fundamental properties of sets. Some identities include identity laws, domination laws, idempotent laws, complementation laws, commutative laws, and associative laws, etc. A table of these identities is provided(see file).
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Examples illustrate how to utilize set identities and notations.
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Description
Explore the fundamentals of sets in discrete mathematics. Understand how to define sets using notation, including examples of standard sets and set builder notation. This quiz will help reinforce your knowledge of basic structures in set theory.
